Generation of time-bin entangled photons without temporal post-selection
Alessandro Rossi, Giuseppe Vallone, Francesco De Martini, Paolo Mataloni
aa r X i v : . [ qu a n t - ph ] M a y Generation of time-bin entangled photons without temporal post-selection
Alessandro Rossi, ∗ Giuseppe Vallone, ∗ Francesco De Martini, ∗ and Paolo Mataloni ∗ Dipartimento di Fisica della “Sapienza” Universit`a di Roma, Roma, 00185 Italy andConsorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, Roma, 00185 Italy (Dated: October 24, 2018)We report on the implementation of a new interferometric scheme that allows the generation ofphoton pairs entangled in the time-energy degree of freedom. This scheme does not require anykind of temporal post-selection on the generated pairs and can be used even with lasers with shortcoherence time.
PACS numbers: 03.67.Bg, 42.50.Dv, 42.65.LmKeywords: time-bin entanglement, spontaneous parametric down conversion
I. INTRODUCTION
Quantum entanglement represents the key ingredientof many quantum information processes. It represents aunique resource that, associated with non-classical corre-lations among separated quantum systems, can be usedto perform computational [1, 2] and cryptographic [3]tasks that are impossible with classical systems. An en-tangled state shared by two or more separated parties isa valuable resource for fundamental quantum communi-cation protocols, such as quantum teleportation [4, 5].In quantum optics, entanglement based on discretevariables is created by the spontaneous parametric downconversion (SPDC) process in a nonlinear (NL) opticalcrystal under excitation of a laser beam, either in con-tinuous wave (CW) or pulse operation. By this processhighly pure entangled states are produced by encodingqubits in a particular degree of freedom (DOF) of thephotons, such as polarization [6], linear and angular mo-mentum [7, 8, 9] and energy-time [10]. More recently, thepossibility of spanning a large Hilbert space by encodingtwo photons in more than one DOF at the same time wasdemonstrated by exploiting the so-called hyperentangle-ment [11, 12, 13, 14]. Energy-time entanglement, alsoincluding the time-bin approach, generally realized withfiber interferometers [15], is based on the interferometricscheme proposed by J. D. Franson [10] which allows thecreation of a superposition state of emission times. Rel-evant realizations of bulk schemes of this idea were alsodemonstrated [16, 17].In this paper we present a novel bulk scheme which al-lows the efficient creation of time-energy entanglementwith SPDC photon pairs emitted by a Type I phasematched NL crystal. At variance with the Franson’sscheme, instead of two phase-locked interferometers (onefor each photon), this apparatus is based on a singleMichelson Interferometer (MI) into which the two pho-tons, travelling along parallel directions, are injected.This configuration reduces the problems of phase insta-bility. Furthermore, by swapping the photon modes in ∗ URL: http://quantumoptics.phys.uniroma1.it/ one of tue two MI arms, the entangled state is auto-matically generated without any need of temporal post-selection. Furthermore, this scheme allows time-energyentanglement to be created with any kind of lasers thatalso have a very short coherence time.The paper is organized as follows. Section II reviewsthe Franson’s unbalanced interferometer and the condi-tions that must be satisfied in order to observe time-binentanglement. In Section III we describe the MI adoptedin our experiment and the generation of the entangledstate . Section IV describes the results of our experi-ment. Finally, in Section V we give the conclusions andindicate some possible use of such scheme.
II. REVIEW OF THE FRANSON’SINTERFEROMETRIC SCHEME
Let’s consider two photons, namely Alice (A) and Bob(B), emitted by a SPDC source towards two differentdirections. Each one is injected into a Mach-Zehnder(MZ) interferometer (cfr. Fig. 1).First order interference arises if the imbalances be-tween the MZ’s long and the short arms∆ x i = ℓ i − s i , i = A, B (1)does not exceed the single photon coherence length, cτ c .We can model the Franson’s scheme as given by two in-dependent sequential operations, corresponding respec-tively to a preparation and a measurement device (seeFig. 1). The first one is realized by the first pair of beamsplitters (BS1 i ) and generates the state | Ψ i = 12 ( | s A i + | ℓ A i )( | s B i + | ℓ B i ) . (2)The detection occurring after the second pair of beamsplitters, BS2 i , projects | Ψ i into the state | Φ i = 12 ( | s A i + e i φ A | ℓ A i )( | s B i + e i φ B | ℓ B i ) . (3)The single counts oscillate as cos φ i because of singlephoton interference and the coincidence rate is expressed (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:8)(cid:10)(cid:11)(cid:12)(cid:7)(cid:4) (cid:1)(cid:2)(cid:3)(cid:4) (cid:5)(cid:6)(cid:7)(cid:8)(cid:2)(cid:9)(cid:6)(cid:3)(cid:6)(cid:10)(cid:11)(cid:1)(cid:9)(cid:6)(cid:4)(cid:7)(cid:9)(cid:7)(cid:11)(cid:12)(cid:13)(cid:10) (cid:1) (cid:1) (cid:2) (cid:1) (cid:2) (cid:2) (cid:1) (cid:2) (cid:14)(cid:15)(cid:16) (cid:1) (cid:14)(cid:15)(cid:16) (cid:2) (cid:14)(cid:15)(cid:17) (cid:1) (cid:14)(cid:15)(cid:17) (cid:2) (cid:3) (cid:1) (cid:3) (cid:2) (cid:18) (cid:1) (cid:18) (cid:2) FIG. 1: Scheme of the Franson’s interferometer. The long andshort arms are labeled as ℓ and s respectively. The thin glasses( φ A , φ B ) can be used to change the phase in the measurement. as C ( φ ) = C cos φ A φ B . (4)In order to create time-bin entanglement the followingconditions must be fulfilled when operating under CWpumping (we consider the two imbalances ∆ x A = ∆ x B ≡ ∆ x equal):i) The arm length difference must be long comparedto the coherence length of photon wavepackets:∆ x > cτ c . (5)This condition, avoiding single photon interfer-ence, is typically satisfied with an imbalance ∆ x ≥ µm , corresponding to detect the photon in thevisible range within a bandwidth of few nanome-ters.ii) The imbalance ∆ x must be lower than the pumpcoherence length τ pump :∆ x < cτ pump . (6)This condition guarantees the coherent superposi-tion of the | s A i| s A i and | ℓ B i| ℓ B i events.iii) ∆ x must be long enough to discard by post-selection the events occurring when one photontakes the short path and the other takes the longpath and vice-versa. These measurement outcomescorrespond to the events | s A , ℓ B i and | ℓ A , s B i . Thislatter requirement imposes the condition∆ x > c ∆ T c , (7)where ∆ T c represents the duration of the coinci-dence window. This condition imposes a strongconstraint on the imbalance ∆ x . Once the above conditions are satisfied, the events | s A , ℓ B i and | ℓ A , s B i are distinguishable and may be dis-carded by a suitable choice of the time coincidence win-dow, while | ℓ A , ℓ B i and | s A , s B i are indistinguishable andgenerates the interference. Indeed, it is impossible to de-termine if the photons are emitted at time t and bothtravel through the long paths or are emitted at time t +∆ x/c and both travel through the short paths. In thisconfiguration, the first beam splitters (BS1 A and BS1 B )generate the state | Ψ i = 12 | ψ ih ψ | + 14 ( | s A , ℓ B ih s A , ℓ B | + | ℓ A , s B ih ℓ A , s B | )(8)where | ψ i = 1 √ | s A , s B i + | ℓ A , ℓ B i ) (9)The second beam splitters (BS2 i ), together with the sub-sequent detection, perform the projection into the entan-gled state | Φ i = 1 √ | s A , s B i + e i( φ A + φ B ) | ℓ A , ℓ B i ) . (10)with phase shifts φ i realized by using two thin glassplates.From equations (6) and (7) it follows that the coher-ence time τ pump of the pump beam must exceed the co-incidence window τ pump > ∆ T c . (11)Typically, the minimum achievable duration of the coin-cidence window is 1 . nsec , hence efficient post-selectioncan be obtained if ∆ x ≥ cm . This generally imposesthe use of a single longitudinal mode laser in order to sat-isfy conditions i), ii) and iii) in the CW regime. Whenoperating with an unbalanced bulk interferometer, anylength variation of the corresponding arms affects thestability of the entangled state, hence this configurationrequires a critical stabilization of the phase. In the caseof an unbalanced interferometer based on single modefibers, the phase is kept stable by by temperature stabi-lization of the fibers. III. THE EXPERIMENT
In the experimental apparatus (see Fig. 2) SPDC pho-ton pairs are generated at degenerate wavelength (wl) λ = 532 nm by a 1 mm slab of β -Barium Borate (BBO),cut for Type I phase matching, excited by a CW singlelongitudinal mode laser (MBD-266, Coherent, coherencetime τ pump > . µsec , λ pump = 266 nm ).In order to inject the two photons into a single MI, thecorresponding k modes must be parallel. On this pur-pose, the characteristic SPDC conical emission of the NLcrystal is transformed into a cylindrical one by a spherical (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:4)(cid:1) (cid:4)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11) (cid:12) (cid:13) (cid:14) (cid:15)(cid:8)(cid:6)(cid:10)(cid:16)(cid:8)(cid:17)(cid:10)(cid:18)(cid:16)(cid:15)(cid:10)(cid:19)(cid:17)(cid:10)(cid:7)(cid:18)(cid:17)(cid:20)(cid:1)(cid:21)(cid:10)(cid:22) (cid:1)(cid:23)(cid:23)(cid:24)(cid:25)(cid:26)(cid:4)(cid:8)(cid:1)(cid:27) (cid:1)(cid:2) (cid:12) (cid:28) (cid:23)(cid:29) (cid:11) (cid:29) (cid:12) (cid:30) (cid:12) (cid:30) (cid:4)(cid:15)(cid:7)(cid:31)(cid:6)(cid:27) (cid:27) (cid:6)!(cid:31)"(cid:1) (cid:23) FIG. 2: Michelson interferometric scheme used to generatetime-bin entanglement. lens ( L P ) whose relative distance from the BBO corre-sponds exactly to its focal length f L = 9 . cm . The lensposition is carefully set by optimizing the visibility of thesingle photon interference. The two photons, travellingalong parallel directions, are injected into the input portof the MI shown in Fig. 2. In this setup the same BSis used both for the preparation of the state in the firstpassage of the photons (corresponding to the BS BS R = T = 0 . ± .
03) beamsplitter (BS). After reflection by mirrors M s and M ℓ , eachphoton experiences a new BS reflection-transmission pro-cess with relative path delay ∆ x = 2( ℓ − s ) . TheBS output beams are then directed by a reflecting prismtowards two single photon counting (SPCM-AQR 14)modules and detected within a bandwidth ∆ λ = 4 . nm ,which corresponds to a single photon coherence time of τ c ∼ f sec . The relative phase of path contributions | s A , s B i and | ℓ A , ℓ B i can be modified by a piezo trans-ducer that finely changes the position of mirror M ℓ onthe ℓ arm. This single MI configuration allows the phaseinstability problems to be minized, as said.The key element of our setup is represented by theswapping operation between the two photons, performedin the ℓ arm of MI. It is realized by the lens L C (see The factor 2 with respect the Franson’s scheme is due to thedouble passage in each arm.
50 100 150 200 2500400080001200016000 0100200300400500 C o i n c i d e n ce s / s ec Piezo Voltage ( x Coincidences V=0.9127–0.0006 S i ng l e s ( x ) / s ec Singles (A)
FIG. 3: Single (right) and coincidence (left) counts measuredby our apparatus in presence of single photon interference.The high visibility of the single count oscillation is used totest the correct position of the lens L P . Coincidence countsalso oscillate in agreement with eq. (4). Figure 2), aligned in such a way that mirror M L is locatedexactly in the focal plane of L C . By this configurationwe can only detect coincidences if both photons travelalong the same arm of the MI. In fact if, for instance,the first photon goes through the ℓ path and the secondfollows the s path (i.e. if we consider the event | ℓ i | s i ),after the second BS passage they will end on the samedetector. Thus no coincidence is detected in this case.The same happens for the | s i | ℓ i event.By this setup it is no longer necessary to perform tem-poral post-selection to discard the events | ℓ i | s i and | s i | ℓ i . They are simply rejected by the coincidencemeasurement. In this way the conditions i) and iii) arenot required anymore. The first one is not necessarybecause single photon interference cannot occur in thisconfiguration. On the other hand, the condition iii) is notnecessary because the events | ℓ i | s i and | s i | ℓ i are notdiscarded by the arrival time, but because of the arrivalposition. This scheme thus allows time-entanglementalso with a very short coherence time of the pump. In-deed time-entanglement is present also if the condition(11) is not satisfied. Moreover the the difference ∆ x canbe very small. IV. EXPERIMENTAL RESULTS
Before describing the experimental results, we show theinterference pattern obtained in the condition of best L P alignment. We used the same setup of Figure 2 with somelittle modifications on the long arm: precisely, we set itslength equal to the short arm and removed the swappinglens L C . In this way single photon interference deter-mines a fringe pattern whose visibility is maximized when
14 15 16 17 18 19 200100020003000 T c =1.5nsecV=0.916–0.027 C o i n c i d e n ce / s ec Piezo Voltage (V)
FIG. 4: Coincidence counts after accidental coincidence sub-traction with ∆ T c = 1 . nsec in the case of time-bin entan-glement. Error bars are calculated by considering both thepoissonian statistic and the uncertainty of the voltage appliedto the piezo. -60 -40 -20 0 20 40010002000300040005000 C o i n c i d e n ce s / s ec Delay (nsec)
FIG. 5: Measurement of a coincidence window ∆ T c =21 . nsec . Fixing the arrival time of the first photon, we mea-sured the coincidences by varing the arrival time delay of thesecon photon. The baseline ( ∼ coinc/sec ) correspondsto the accidental coincidence contribution. This value is inagreement with that obtained by equation (13). the two photons’ directions are exactly parallel. This isobtained when the distance between L P and the BBOcrystal is equal to the lens focal lenght. The position of L P is then carefully set by maximizing the interference.The corresponding results are shown in Fig. 3.After optimization of lens L P we set the measurementapparatus in the configuration of the Figure 2. Here theimbalance is ∆ x = 120 cm and we do not expect anysingle count oscillation, while time-bin entanglement willarise. Fig. 4 shows the coincidence rate as a function ofthe voltage applied to the piezo for a coincidence win- T c =21.5nsecV=0.664–0.008 C o i n c i d e n ce s / s ec Piezo Voltage (V)
Without accidentalsubtraction
FIG. 6: Raw coincidence counts (without accidental coinci-dence subtraction) with coincidence window ∆ T c = 21 . nsec . Piezo Voltage (V)V=0.949–0.005 C o i n c i d e n ce s / s ec T c =21.5nsec With accidentalsubtraction
FIG. 7: Coincidence counts with 21 . nsec coincidence windowafter accidental subtraction. dow ∆ T c = 1 . nsec . Each datum is obtained in 1 sec acquisition time. As usual, the visibility is defined as V = C max − C min C max + C min (12)where C max(min) is the maximum (minimum) coincidencevalue in the oscillation pattern. Equivalently, it is de-fined as the parameter V in the fitting function C ( x ) = c (1+ V cos[ ω ( x − x )]), with x corresponding to the piezovoltage. The high visibility value of the V = 0 . ± . | s A , ℓ B i and | ℓ A , s B i were not discarded thevisibility could not be larger than 0 .
5. It is worth not-ing that the small value of ∆ T c would allow these eventsto be discarded even in the case of a standard Franson’sscheme.Let’s now describe the case of a much longer coin- (cid:1) (cid:2) (cid:1) (cid:2)(cid:2)(cid:3) (cid:3)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:9) (cid:14)(cid:15)(cid:9)(cid:10)(cid:16)(cid:17)(cid:15)(cid:11)(cid:18)(cid:8)(cid:19)(cid:5)(cid:20)(cid:15)(cid:21)(cid:5)(cid:12)(cid:15)(cid:8)(cid:11) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8) (cid:14)(cid:15)(cid:9) (cid:1)(cid:2)(cid:3)(cid:4) (cid:2)(cid:22) FIG. 8: Scheme for the generation of two photon polarization/momentum/time-bin entangled states. cidence window, ∆ T c = 21 . nsec . The value of ∆ T c was measured by observing the coincidence pattern as afunction of the relative temporal delay between the twophotons (see Fig. 5). Precisely, we fix the arrival timeof the first photon and measure the coincidence num-ber by varying the arrival time δt of the second photon.Since the coincidence box detects coincidences regardlessof which photon comes first, we detect real coincidencesif − . nsec ≤ δt ≤ . nsec . In this condition, by us-ing the standard Franson’s scheme the imbalance ∆ x would not allow time-bin entanglement because the con-dition (7) would be violated. In our scheme equation (7)is not necessary anymore; here we can generate time-binentanglement even if the imbalance ∆ x is shorter thanthe coincidence window ∆ T c . In Fig. 6 we show thecoincidence pattern obtained with ∆ T c = 21 . nsec . Wereport here the raw data, i.e. without any accidental co-incidence subtraction. As expected, even if the visibilityis quite low V = 0 . ± . .
5. In this con-dition the standard Franson’s scheme would not allowa visibility > .
5, due to the impossibility of discardingthe events | s A , ℓ B i and | ℓ A , s B i . We also report in Fig.7 the experimental results of Fig. 6 after accidental co-incidence subtraction. These are estimated by using theexpression C acc = 2 S A S B ∆ T c , (13)where S A and S B are the single counts measured at de-tectors D A and D B respectively. The new visibility isnow V = 0 . ± .
005 which is comparable within theerrors with that obtained with ∆ T c = 1 . nsec . V. CONCLUSIONS
In this paper we have described a novel interferomet-ric scheme to create time-bin entanglement based on twophoton states. It consists of a Michelson interferome-ter in which two degenerate photons, created by spon-taneous parametric down conversion in a Type I phase matched NL crystal and travelling along parallel direc-tions, are injected. Compared with the standard Fran-son’s scheme, which adopts two different unbalanced in-terferometers and creates time-bin entanglement by tem-poral post-selection, our scheme presents significant ad-vantages: • Since it is based on a common interferometer forthe two photons, it avoids most of the problemsrelated to phase instabilities. • The scheme avoids the use of temporal post-selection to generate the entanglement. Indeed,by using a simple lens in one of the interferome-ter arms, it operates under the spatial swapping ofthe two photon k -modes. • Under CW operation, this scheme can work withany kind of laser (even with a short coherence time)used to pump the parametric process.The experimental results obtained by this scheme indi-cate that it represents a possible solution for several ap-plications of quantum communication and quantum com-putation. In particular, it can be used to increase the di-mension of a multi-degree of freedom two photon state.For instance, it can be used to add two more qubits bytime-energy entanglement to a four qubit polarization-momentum hyperentangled state [18, 19]. By referringto the parametric source sketched in Figure 8, at the be-ginning, polarization entanglement is generated by dou-ble passage of the pump beam and of the photon pairthrough a Type I BBO crystal, and a λ/ k modes ofthe SPDC conical emission of the Type I crystal. Inthis way polarization-momentum hyperentanglement isgenerated (see [12] for a detailed description of the hy-perentangled source). This state corresponds to a two-photon four-qubit state. By injecting the four modesinto the MI described in this paper we will add time-bin entanglement and allow each photon to encode threequbits in the different DOF’s. This will generate thepolarization/momentum/time-bin entanglement of twophotons. This experiment is at the moment under in-vestigation. Acknowledgments
Thanks to Gaia Donati for the contribution given inrealizing the experiment. This work was supported by the PRIN 2005 of MIUR (Ministero dell’Universi`a e dellaRicerca), Italy. [1] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. ,5188 (2001).[2] E. Knill, R. Laflamme, and G. J. Milburn, Nature ,46 (2001).[3] A. K. Ekert, Phys. Rev. Lett. , 661 (1991).[4] D. Boschi, S. Branca, F. De Martini, L. Hardy, andS. Popescu, Phys. Rev. Lett. , 1121 (1998).[5] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. We-infurter, and A. Zeilinger, Nature , 575 (1997).[6] P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V.Sergienko, and Y. Shih, Phys. Rev. Lett. , 4337 (1995).[7] J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. , 2495(1990).[8] N. K. Langford, R. B. Dalton, M. D. Harvey, J. L.O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, andA. G. White, Phys. Rev. Lett. , 053601 (2004).[9] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature , 313 (2001).[10] J. D. Franson, Phys. Rev. Lett. , 2205 (1989).[11] C. Cinelli, M. Barbieri, R. Perris, P. Mataloni, and F. De Martini, Phys. Rev. Lett. , 240405 (2005).[12] M. Barbieri, C. Cinelli, P. Mataloni, and F. De Martini,Phys. Rev. A , 052110 (2005).[13] T. Yang, Q. Zhang, J. Zhang, J. Yin, Z. Zhao,M. Zukowski, Z.-B. Chen, and J.-W. Pan, Phys. Rev.Lett. , 240406 (2005).[14] J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G.Kwiat, Phys. Rev. Lett. , 260501 (2005).[15] J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, Phys.Rev. Lett. , 2594 (1999).[16] P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, Phys.Rev. A , R2472 (1993).[17] P. R. Tapster, J. G. Rarity, and P. C. M. Owens, Phys.Rev. Lett. , 1923 (1994).[18] G. Vallone, E. Pomarico, F. De Martini, and P. Mataloni,Phys. Rev. Lett. , 160502 (2008).[19] G. Vallone, E. Pomarico, F. De Martini, and P. Mataloni,Laser Phys. Lett.5